Chronicle

ChronicleSource: SIAM Review, Vol. 14, No. 3 (Jul., 1972), pp. 520-546Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2028424 .

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SIAM REVIEW

Vol. 14, No. 3, July 1972

CHRONICLE

All communications should be sent to H. B. Hair, SIAM Publications, 33 S. 17th Street, Philadelphia, Pa. 19103.

SIAM 1971 Fall Meeting

The Society for Industrial and Applied Mathematics held its Fall Meeting at The University of Wisconsin, Madison, Wisconsin, on October 11, 12 and 13, 1971. Total registration was 250 including the following 137 members of SIAM:

R. D. Adams, University of Kansas; Y. Alavi, Western Michigan University; E. Allgower, Colorado State University; P. M. Anselone, Oregon State University; K. E. Atkinson, Indiana University; W. W. Babcock, Northern Michigan University; V. Barcilon, University of California, Los Angeles; E. H. Bareiss, Argonne National Laboratory; E. R. Barnes, University of Wisconsin; C. F. A. Beaumont, University of Waterloo; R. Beckett, U.S. Army Weapons Command; P. R. Bender, Marquette University; T. R. Benedict, Cornell Aero Laboratory; B. Bernstein, Illinois Institute of Technology; N. Bleistein, University of Denver; I. E. Block, AUERBACH Info., Inc.; N. N. Bojarski, Physicist, Consultant; W. R. Boland, Clemson University; W. M. Boyce, Bell Telephone Laboratories Inc.; S. L. Brodsky, Office of Naval Research; G. D. Byrne, University of Pittsburgh; B. G. Carlson, Los Alamos Scientific Laboratory; R. E. Carr, University of Wyoming; J. Case, Johns Hopkins University; E. W. Chapin, Jr., University of Notre Dame; B. A. Chartres, University of Virginia; J. S. Chipman, University of Minnesota; A. K. Cline, National Center for Atmospheric Research; J. A. Cochran, Bell Telephone Laboratories; J. D. Cole, University of California, Los Angeles; B. H. Colvin, Boeing Scientific Research Laboratories; C. Crown, Purdue University; J. T. Day, Penn State University; C. A. Deavours, The Cooper Union; D. W. DeMichele, Texas A. & M. University; R. C. DiPrima, Rensselaer Polytechnic Institute; H. E. Fleming, National Oceanic and Atmospheric Administration; E. Frank, University of Illinois, Chicago Circle; David Gale, University of California, Berkeley; D. M. Girard, University of Wisconsin, Green Bay; J. W. Givens, Argonne National Laboratory; J. Goebel, Montana College of Mineral Science-Technology; K. Gopalsamy, University of Calgary.

J. Haley, Gulf Research and Development Company; R. Handelsman, University of Illinois, Chicago Circle; R. J. Hanson, California Institute of Technology; E. D. Harrison, Michigan State University; D. Hector, University of Denver; H. Hethcote, University of Iowa; W. C. Hoffman, Oakland University; B. E. Howard, University of Miami; G. C. Hsiao, University of Delaware; Y. Ikebe, University of Texas; W. J. Jameson, Jr., Collins Radio Company; M. Jeppson, Colorado State University; L. H. Jones, University of Delaware; W. J. Kammerer, Georgia Institute of Tech- nology; H. Kaper, Argonne National Laboratory; C. L. Keller, U.S.A.F.; J. Keller, Monsanto Research Corporation; P. G. Kirmser, Kansas State University; R. E. Kleinman, University of Delaware; T. C. Koopmans, Yale University; J. B. Kruskal, Bell Telephone Laboratories; R. G. Lamb, AUERBACH Associates, Inc.; L. J. Lardy, Syracuse University; R. Lau, Office of Naval Research; J. S. Lew, IBM Corporation; C. C. Lin, Massachusetts Institute of Technology; P. Linz, University of California, Davis; V. Lovass-Nagy, Clarkson College of Technology; 0. L. Mangasarian, University of Wisconsin; C. G. Maple, Iowa State University; D. Matula, Washington University; J. McKenna, Bell Telephone Laboratories; R. E. Meyer, University of Wisconsin; R. R. Meyer, Shell Development Company; L. W. McKenzie, University of Rochester; R. Miller, Iowa State University; M. Minkoff, University of Wisconsin; C. Moler, University of Michigan; J. J. More, Cornell University; J. A. Morrison, Bell Telephone Laboratories; Z. C. Motteler, Gonzaga University; M. N. L. Narasimhan, Oregon State University; N. Z. Nashed, University of Wisconsin; R. W. Nau, Carleton College; P. Nelson, Oak Ridge National Laboratory; E. N. Oberg, University of Iowa.

S. V. Parter, University of Wisconsin; M. L. Patrick, Duke University; M. Pavel-Givens, Sorbonne; L. E. Payne, Cornell University; A. L. Perrie, Wisconsin State University; J. L. Phillips, Washington State University; D. T. Piele, University of Wisconsin, Parkside; W. G. Poole, Jr., College of William and Mary; R. D. Prasad, IBM Corporation; D. A. Prelewicz, Washington University; L. B. Rall, University of Wisconsin; S. W. Rauch, University of New Brunswick; A. K. Ray, University of

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Ottawa; G. W. Reddien, Vanderbilt University; J. Reichbach, Technological University, Holland; V. C. Rideout, University of Wisconsin; E. L. Roetman, University of Missouri, Columbia; F. V. Rohde, Mississippi State University; R. Saeks, University of Notre Dame; D. F. Sanderson, Western Washington State College; J. P. San Giovanni, Esso Mathematics and Systems, Inc.; P. E. Saylor, University of Illinois, Urbana; H. E. Scarf, Yale University; D. H. Schultz, University of Wisconsin, Milwaukee; H. K. Schultz, Wisconsin State University; M. Scott, Sandia Laboratories; C. B. Shaw, Jr., North American Rockwell; R. F. Sincovec, Kansas State University; K. Singh, Saint Mary's University; J. L. Snell, Dartmouth College; N. Stahl, University of Wisconsin, Green Bay; R. S. Stepleman, University of Virginia; J. Stover, Memphis State University; 0. N. Strand, National Oceanic and Atmospheric Administration; D. D. Sutherland, Babcock and Wilcox; L. Tavernini, Arizona State University; V. Thom&e, University of Wisconsin; D. L. Thomsen, Jr., IBM Corporation; T. Triffet, Michigan State University; B. L. Turlington, Southern Methodist University; V. Vidyasagar, Sir George Williams University; G. Wahba, University of Wisconsin; G. I. Wakoff, A. T. & T.; W. R. Wasow, University of Wisconsin; D. S. Watanabe, University of Illinois; R. P. Weber, Bell Telephone Laboratories; R. Wernick, SUNY, Oswego; E. Wolman, Bell Telephone Laboratories; K. Wright, University of Toronto, Canada; A. H. Zemanian, SUNY, Stony Brook.

Titles and abstracts of papers presented are as follows.

G. D. ANDRIA, G. D. BYRNE and C. A. HALL, University of Pittsburgh. Convergence of Cubic Spline Interpolants of Functions Possessing Discontinuities.

Let SJf denote the cubic spline interpolant on I = [-1, 1] to f corresponding to the mesh 7tn

whose mesh gauge is 6(. If the restrictions on 7tn are the same as those used in [C. A. Hall, Uniform convergence of cubic

spline interpolants, J. Approx. Theory, to appear], then we establish pointwise convergence of {(SJf)(x)} tof(x) forf E C(I - {}),f bounded on I, and x 0 0 as -- 0. We also establish analogous results for a finite number of discontinuities and for f Lipschitz continuous except at a finite number of points.

Next, if the only restriction on 7tn is that 6(5 - 0 as n - oo, we show that SJf converges uniformly to f if f is a continuous function on I whose first derivative is piecewise continuous.

Finally, some examples are given.

K. E. ATKINSON, Indiana University. An Outline of Numerical Methods for Fredholm Integral Equations of the Second Kind. (Invited.)

An outline is given for the numerical analysis of Fredholm integral equations of the second kind; methods of general applicability are emphasized. The topic headings are as follows. 1. Successive approximation and its generalizations. 2. Degenerate kernel methods and projection methods (e.g., collocation, Galerkin's method). 3. The Nystrom method, or equivalently, the procedure of replacing the integral operator by a numerical integral operator in order to approximate the integral equation by a finite linear analogue. 4. Iterative variants of the direct methods; these are iterative methods for solving the linear systems arising in categories 2 and 3. 5. Automatic computer routines for integral equations; the problems of writing such routines, along with some possible solutions.

VICTOR BARCILON, University of California, Los Angeles. On a Nonlinear Integral Equation. A class of optimization problems arising in the design of electrical filters and antennas is reduced

to the solution of a nonlinear integral equation analogous to the Chandra-Sekhar equation in radiation theory. The problem consists in "confining" both a function and its Fourier transform subject to additional constraints. Two particular examples are examined to illustrate the general theory.

EDWIN H. BAREISS, Argonne National Laboratory and Northwestern University. Numerical Solution of the Transport Equation. (Invited.)

This lecture concentrates on the numerical solution of the neutron transport equation. Because of the current need for research and development of fast nuclear power reactors, as well as public concern on nuclear reactor safety, calculations based on transport theory become more and more important in two and three space dimensions, as opposed to the more traditional calculations based on diffusion theory.



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A short introduction to the neutron transport equation will be given, and the main problems connected with it will be pointed out. The conclusion is that highly efficient algorithms as well as high order approximations are needed even when the most advanced computers are used. So far, the reactor engineers and mathematicians have devised ingenious methods to break huge problems down into sets of manageable smaller problems. This is achieved for the price of reduced reliability of the computed data. The mathematical reason for the computational difficulties are in the nonself-adjointness of the transport operator. Therefore, a precise understanding of the analytic behavior of the solutions of the transport equation is necessary.

On a simplified model, it is shown how the transport operator can be transformed into other operators which reveal some of its specific properties. Special attention will be given to the behavior of solutions at boundaries and interfaces. Neglect of this behavior can pose serious difficulties in high order approximation techniques. It will be shown how so-called "benchmark problems" can be constructed and used to ensure the proper functioning of a code.

A number of benchmark problems have been solved at Argonne National Laboratory and Northwestern University using finite difference and variational techniques for different degrees of approximations. The numerical results will be graphically displayed and summarized and some of the "strange" results will be explained.

BARRY BERNSTEIN, HERBERT WEINSTEIN, Illinois Institute of Technology, and AARON B. SHAFFER,

Michael Reese Hospital, Chicago. Volterra Integral Equations in Blood Circulation Measurements. Whole body circulating blood volume and other related quantities may be deduced from tracer

dye measurements on the circulation. However, a measured dye curve results from a superposition of dye making its first circulation, dye making its second circulation, third circulation, etc. It is shown that the primary circulation curve P(t), which is the distribution of first circulation times, is related to the normalized measured dye curve Q(t), by the volterra integral equation

Q = P + Q * P,

where * denotes convolution. This equation is solved numerically by a Liouville-Newmann series method, which converges rapidly since it can be shown that in any finite interval of time all but a finite number of terms of the series vanish. This follows from the existence of a finite appearance time; i.e., there is an a such that Q(t) = 0 for t < a. Correction for the distortion of the sampling system involves solving for Q a Volterra integral equation of the form

F = G *Q,

where F and G are known functions. A solution is obtained by a Fourier transform method. Formulas for the moments of Q in terms of those of F and G are obtained through Laplace transform methods. The method is applied to human and canine data.

JEAN-MICHEL BISMUT, Ingenieurau Corps des Mines, IRIA, Rocquencourt, France. Some Extensions of Team Theory.

Let (A, a, 1) be a measured space, representing the set of agents. We suppose that a is a separable sigma algebra. (For the probabilistic definitions, see P. A. Meyer, Probabilite et potentiels, Hermann, Paris.) Let (0, o, P) be a probability space. We call structure of information-action a subsigma algebra of a 0 o.

Example. If for each x in A, we define a sigma algebra b(x) in o, we can associate the following structure: it consists of the measurable sets of A x 0, I such that 1I(x, *) is b(x) measurable for each x. We suppose that a = U an, the an being an increasing sequence of countable partitions of A. If an = (a', a ,), let a,x) be the only set of the partition an such that xne a(X). We define b(aX) as U yb(y), y E a,. If b(x) = n b(a,nX)), for X in L1(A x 0), the conditional expectation of X relative to the previous structure is the class of the measurable versions of E(X(x, * )/b(x)). For instance, A = [0, 1], b(x) is an increasing right-continuous sequence of sigma algebras.

THEOREM. Let I be an information-action structure, V a reflexive Banach space, which is supposed to be separable. Let j(y, * ) be a family of convex functionals on Lp(A; V)(1 < p < + cx) such that:

(i) For X in LP(A; V),j(y, X) is measurable on 0.



CHRONICLE 523

(ii) One canfind k1, k2 > O,f1 ,f2 in L1(O) such that k, IIXIILP(A;V) + f1(y) _ j(y, X) _ k211XIILp(A;V)

+ f2(y). Let Lp(I; V) be the closed subspace of LP(A x 0; V) of I-measurable functions. Then the funictionial E(j(., X( . ))) has a miniimum oni LP(I; V). A characterizationi of a miniimum X? is: One can find Y measurable on A x 0, with values in V' such that Y(y, ) E j(y, X0), with E(Y/I) = 0. Moreover Ye Lp,(A x O; V').

COROLLARY. Let II , In be n iniformationi-actioni structures, J1 ... Jn be a partitioni of 0 such that A x J, is I, measurable. Let I be the following structure: It consists of the subsets of A x 0 such that their intersection with A x J, is Ii measurable. Then X? can be written as 'A x X?X, X? minimizing the functional relative to the structure Ii.

This is the main result for studying the exception procedures as in R. Radner's Evaluation of information in organisations [Proc. 4th Berkeley Symposium on Mathematical Statistics and Probability, Univ. of California Press, Berkeley, 1961].

W. ROBERT BOLAND, Clemson University. The Numerical Solution of Fredholm Integral Equations Using Product Type Quadrature Formulas.

A product type quadrature formula for the integral f'f(x)g(x) dx is the bilinear form fTAg, where A is an (n + 1) x (m + 1) matrix of constants, f = (f(x0),f(x) , f(xn))' and g = (g(yo),g(y1), * , g(ym))T. The x's and y's are respectively distinct real numbers in the domains of f and g. The integral in a Fredholm integral equation is approximated by a product-type quadrature formula. The solution of the approximating equation is obtained by solving a related linear algebraic system. A convergence theorem is established using Anselone and Moore's general theory of linear operators. Several numerical examples are included.

WILLIAM M. BOYCE, Bell Telephone Laboratories, Inc., Murray Hill, N.J. A Modified Random Walk Modelfor Speculative Prices.

The Gaussian random walk hypothesis for speculative prices may be combined with subjective probabilities for future prices to produce an interesting stochastic process. Early results on this process were presented in the author's Stopping Rules for Selling Bonds [Bell. J. Econ. Management Sci., 1 (1970), pp. 27-53]. Here additional properties of the process are described along with applications to various problems of financial optimization.

AMNON BRACHA, University of Illinois. A Symmetric Factorization Procedurefor the Solution of Elliptic Boundary Value Problems.

The convergence properties of a symmetric factorization procedure for solving elliptic difference equations of the type Au = q, is discussed.

The idea of a factorization is to find a matrix A + B that can be factored as the product of sparse matrices L and U,

A + B = LU.

The efficient solution of (A + B)U = q is then used on the iterative scheme defined by

(A + B)Ui+1 = (A + B)U, - z(AUi -q),

where z is a parameter. Convergence of the method depends on bounds on the parameter z. Computable values of these bounds are obtained. The procedure and the results are generalized to an arbitrary number of dimensions.

JAMES CASE, Johns Hopkins University. On the Form of Market Demand Functions. Consider a market in which N firms, called 1, 2, *.. , N, compete to sell a single good for which

the total demand is highly inelastic. Let p, be the price charged by firm i for brand i of this good, and let Di(p1, , PN) denote the market demand for brand i. The functions D1, , DN should possess the following four properties:

(i) Each Di should depend on the price ratios P/P1, , P/PN alone. (ii) D = ,, Di(pl, * *, PN) should (because of the inelasticity of total demand) be a constant

independent of prices. (iii) If p, = 0 and p, > 0 for each j 0 i, then Di should equal D.



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(iv) If p, = X, then D, should vanish and the functions DJ,j : i, should describe a similar market of only N - I competitors.

We shall display a class of demand functions D, possessing the properties (i)-(iv), discuss what little empirical evidence exists in their behalf, and draw a few elementary conclusions about markets governed by such demand functions.

E. WILLIAM CHAPIN, JR., University of Notre Dame. On the Complexity of Automatal Computationt. When both the generation of a language by a grammar and the computation performed by a

nondeterministic automaton are viewed as trees of a certain type, the complexity of the language or the computation can be measured, not in terms of time and tape bounds, but in terms of the complexity of the rate of branching of the tree in question. A suitable measure of complexity of this type is constructed and is then applied to the formal description of programming languages.

B. A. CHARTRES and R. S. STEPLEMAN, University of Virginia. Convergence of Difference Methods for Initial Boundary Value Problems with Discontinuous Data.

This paper extends the classical convergence theory for numerical solutions to initial and boundary value problems with continuous data (the right-hand side) to problems with Riemann integrable data. Order of convergence results are also obtained. A representative theorem is the following one.

THEOREM. Suppose f :R2 -- R' is Riemann integrable along the solution curve to y'(t) = f(t, y(t)), y(O) = a; and f(t, a) is Lipschitz continuous in the second variable. Then Euler's method converges uniformly to the exact solution on [0, 1]. If, in addition, f is of bounded variation along the solution curve, then the convergence is of order h.

JOHN S. CHIPMAN, University of Minnesota. A Renewal Model of Economic Growth. Let a return function a and a depreciable rule 6 be integrable real-valued functions defined over

0 < t < x. corresponding to investment of one unit at t = 0. Assume that investment of i units at t = to yields flows of Xa(t - to), Xb(t - to) over a period to < t < cc. Gross output x and depreciation expense u are determined by

x(t) f a(z)v(t - z) dz, u(t) = f 6(z)v(t - z) dz, 0 0

where v is gross investment, assumed determined by v(t) = u(t) + sy(t), 0 < s < 1, where x = y - u is net national income, which is found to obey the renewal equation

y(t) = g(t) + f f (l)y(t - z) dzl,

where f (t) = sa(t) + (1 - s)b(t), g(t) = St f (z)y(t - z) dz. From Feller's results we have

y(t) - 7/7r e rIt as t -- >, where r, is the largest positive root of Sf e -f (t) dt = 1 = St e01g(t) dt, and m = S0 e-1tf(t) dt is the "period of production."

Result 1. Let a(t) > 0, S a(t) dIt > 1. Then the Harrod-Domar formula r1 = ks holds for some constant k if and only if 6 is the Hotelling declining-value policy

6(t) = a(t) - Pw(t), w(t) - e(t-r)a(T) d.

Moreover, k = P where P is the internal rate of return (largest root of S0 e-rta(t) dt = 1). Result 2. Let a E A satisfy the above with a slowly varying at infinity and f0 ta(t) dt < c; let

Sf ' tg(t) dt < oc; and let 6 be as above with 6(t) _ 0. Then given a, a' E A with rates of return P, P' and periods of production 7t, 7r', the "overtaking principle"

y >- y' < (3TO)(VT > To) f [y(t) - y'(t)] dt > 0 0

induces an ordering of A representable by the lexicographic order

a a'. P > P' or P = P' and r < r'.



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Result 1 is established by Laplace transform techniques, and Result 2 by means of Wiener's general Tauberian theorem.

A. K. CLINE, National Center for Atmospheric Research, Boulder, Colo., and University of Colorado. Two Descent Methods/for Determining the Chebyshev Solution to an Overdetermined System of Linear Equations.

Given an in x n matrix A (m > n), and an in-vector b, we seek to determine an n-vector x to minimize the uniform norm of the residual r = Ax - b. Two very similar methods will be presented for solving this problem. Both techniques have the following characteristics:

1. The solution is determined in a finite number of steps. 2. Strict descent to the solution (i.e., at each step the norm of the residual at the present approximate

solution is strictly less than the norm of the previous residual). 3. n x n linear sybsystems are solved at each iteration. Other methods often use (n + 1) x (n + 1)

subsystems. 4. The sequence of subsystems has the property that one subsystem differs from the preceding

system in only several rows. L U decomposition is optimized using this property. Numerical experience has shown these techniques to be very efficient for very "thin" systems

(m >> 1). A short computer produced film will be shown describing geometrically the algorithms.

J. A. COCHRAN and E. W. HINDS, Bell Telephone Laboratories, Inc., Whippany, N.J. Numerical Determination of Partial Eigensystems of the Complex Symmetric Kernels of Laser Theory.

This paper is devoted to the application of certain quadrature techniques to the solution of the partial eigenvalue problem for a class of integral equations arising in laser theory. The kernels of these equations are, in general, complex symmetric, and one is usually interested in determining the first several lower order eigenvalues and eigenfunctions. By exploiting certain properties of the kernels in question, a rapid accurate procedure has been developed which appears to have several distinct advantages over techniques previously utilized in related investigations. Moreover, application of the method to several cases of interest not only has provided results in good agreement with those of earlier researchers, but also has extended considerably the theoretical knowledge of the modal configurations associated with the allied optical structures. Typical results will be given for the complex- symmetric kernel K(x, y) = exp [- ik(x - y)2], where k is a real parameter. The beneficial performance of the general approach in this and related examples suggests that broader applicability may lead to similar advantages in other practical problems.

J. C. CROWN, Indiana University-Purdue University at Indianapolis. Stable Accurate Single- Evaluation Predictor-Corrector Algorithmsfor the Numerical Solution of Ordinary DifferentialEquations.

Runge-Kutta type algorithms for the numerical solution of ordinary differential equations are highly accurate but require several derivative evaluations per step. The usual predictor-corrector methods require only two derivative evaluations per step but for the same order are not as accurate as the Runge-Kutta type methods. Predictor-alone methods require only one derivative evaluation per step but are less accurate and less stable than corresponding predictor-corrector methods. A class of predictor-corrector methods requiring only one derivative evaluation per step has been developed. These new algorithms are not only stable but more accurate than corresponding double-evaluation predictor-corrector methods. The theory of these new algorithms is presented together with cor- roborating numerical results.

C. A. DEAVOURS, The Cooper Union for the Advancement of Art and Science. A Singular Expan- sion Problem Arising from Forced Separation of Variables.

Frequently, certain PDE's are "forced" to separate in order to construct eigenfunction series solutions to boundary value problems. For example, the biharmonic equation AAT = 0 is nonseparable but has solutions of the form Tl = X(x)exp(-xcy) if X(x) satisfies X('v) + 2xC2X" + Xt4X = 0. To satisfy boundary data, eigenfunction expansion methods are necessary for the equations obtained which always involve eigenvalue parameters in a nonlinear manner. Most of the ordinary differential equations encountered can be converted to equivalent vector systems in which the eigenvalue depend- ence is linear. Elliptic equations generally lead to vector systems in Hilbert space for which spectral theory is well-developed. Hyperbolic equations involve systems in linear spaces with indefinite bilinear



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forms and the necessary spectral questions have not yet been fully answered. The author briefly reviews three prototype nonseparable equations encountered in practice for which the above problems have been solved. A singular expansion problem derived from a hyperbolic equation is then investigated. The resulting vector system has a continuous spectrum consisting of two linear line segments in the complex c-plane and furnishes a generalization of limit point behavior of the classical case. Proofs are indicated to the above problem.

LOKENATH DEBNATH, East Carolina University. On an Unsteady Motion of a Rotating Stratified Fluid.

Based on the Boussinesq approximation, a theory of an unsteady flow induced in an unbounded rotating stratified fluid by an impulsive motion of a sphere which starts oscillating along the axis of rotation of the fluid with a given velocity is presented. The properties of the resulting flow associated with relative magnitude of the forcing frequency co of the sphere, the Brunt-Vaisala frequency N and the angular velocity Q of the fluid have been investigated. The velocity field related to these frequencies and the various regions of the space has been obtained and its characteristic features have been explored in the rotating and/or stratified fluid. It has been shown that the inertial-internal waves are likely to occur which carry the transient energy at a frequency between 2Q and N. The wave-like nature of the motion is entirely lost provided w lies outside of the domain 20 to N. The pattern of the flow in the far field and at large time has been examined. It has been shown that Stewartson's [Proc. Cambridge Philos. Soc., 48 (1952), p. 168] and Mallick's [Proc. Nat. Inst. Sci. India Part A, 23 (1957), p. 544] results can be retrieved as special cases. Several interesting features of the flow have been investigated.

D. W. DEMICHELE, Texas A. & M. University, and M. V. DAVIS, University of Arizona. A Numerical Solution to a Laminar Axial Symmetric Compressible Flow Problem with "Blowing" and "Suction."

The vapor flow within a cylindrical heat pipe can be described as a compressible flow problem with "blowing" and "suction." The velocity regime of the vapor extends from very low speeds to supersonic velocities. This work presents the development of a numerical method to solve two- dimensional symmetric flow problems of flowing gases or vapors with or without viscous and heat transfer effects. The problem has been solved by use of an integral transformation of the general equations of continuity, momentum and energy. The integral transformation is closely related to the stream function transformation first introduced by R. Von Mises. The method converts the general two- dimensional multivariable set of elliptic partial differential equations into a set of integral differential equations defined on a set of nonorthogonal curvilinear coordinates (stream tubes). In the supersonic regime, two unique solutions have been found. In order to verify the method and its solutions, closed form solutions applicable to problems with very low mach numbers have been compared with the results of the numerical solution and show excellent agreement. For the high mach number regime, comparisons between experiments and the numerical results have been made again with good agreement. The method has been found to be both numerically efficient and stable.

J. E. DENNIS, JR., Cornell University, J. F. TRAUB, Carnegie-Mellon University, and R. P. WEBER, Bell Telephone Laboratories, Inc., Whippany, N.J. On the Matrix Polynomial and Lambda-Matrix Problems.

A matrix S is a solvent of the matrix polynomial

M(X)- Xm + AlXm-l + + Am,

if M(s) = 0, where the A,'s, X and S are n x n matrices. It is shown that there exist matrix polynomials with no solvents. However, if the scalar polynomial det M(MI), where A is a scalar, has mn distinct roots, then M(X) has m solvents, S1, , Sm, with distinct and disjoint eigenvalues. Furthermore, the block Vandemonde of these solvents,

V(S1, ,Sm) (S,- 1) for i,j = 1,.,

is nonsingular. Note that in general V(X1, *, Xm) may be singular even if X1, l , Xm have distinct and disjoint eigenvalues.

An algorithm is presented that is based on Traub's scalar polynomial algorithms. The algorithm yields a solvent S, if:



rTJPCXNTTrI .F 527

(i) M(X) has m solvents, S1, , Sm, (ii) the eigenvalues of S, dominate those of Si for i # 1, (iii) V(S1, , Sm) and V(S2, , Sm) are nonsingular.

Numerical results are given.

M. A. DIAMOND, University of Illinois. Iterative Solution of Implicit Approximations of Partial Differential Equations.

An algorithm is developed from factorization methods to solve the large sets of algebraic equations that arise in the approximate solution of partial differential equations by implicit numerical techniques. Factorization methods are iterative, and a knowledge of the largest and smallest eigenvalues of the iteration matrix is needed to determine a set of parameters used in the iteration. An advantage of the algorithm over those now in use is that no knowledge of the eigenvalues is required to start the iteration. It is shown that the factorization iteration can be used to optimally compute the required eigenvalues. Then the algorithm is developed to solve the set of simultaneous linear equations and dynamically improve the set of parameters by generating successively better approximations to the eigenvalues of the iteration matrix. The approximations are computed from quantities calculated by the factorization iteration. It is shown that the algorithm developed is always convergent.

R. C. DIPRIMA, Rensselaer Polytechnic Institute, and J. T. STUART, Imperial College, London. Nonlocal Effects in the Stability of Flow between Eccentric Rotating Cylinders.

In this paper the linear stability of the flow between two long eccentric rotating circular cylinders is considered. This problem is of interest in lubricating technology. The basic flow has components in the radial and azimuthal directions and depends on both these coordinates; hence, the stability equations are partial differential equations rather than ordinary differential equations. Thus standard methods of hydrodynamic stability theory are not applicable. There are two small parameters, the clearance ratio and the eccentricity. By letting these two parameters tend to zero in an appropriate manner, a global solution to the stability problem is obtained by essentially the method of multiple scales without recourse to the concept of "local instability" or "parallel-flow" approximation, so commonly used in boundary-layer stability. The results, which are at variance with those obtained by a local theory, are in good agreement with experimental observations for small clearance ratios and small eccentricity.

M. DUDUKOVIC, H. WEINSTEIN, B. BERNSTEIN, Illinois Institute of Technology, and A. B. SHAFFER,

Michael Reese Hospital, Chicago. Mathematical Models for Capillary Exchange. A system of coupled partial differential equations with appropriate boundary and initial conditions

is used to describe the unsteady solute transport between blood plasma in a single capillary and adjacent tissue space. Separation of variables and standard transform techniques failed to yield a closed form solution due to the coupling of a wave equation with a diffusion equation through a common boundary condition. The solution to the general model is presented in the form of an integral equation which can be solved by successive iterations. This integral equation in the Laplace transform plane reduces to a Fredholm equation with a separable kernel. Convergence of the solution and the effort required to compute the results are discussed. As the kernel is represented by a double infinite series, numerical computations of successive approximations are tedious and lengthy. However, for certain values of system parameters the iteration scheme becomes applicable. Several simplified models are derived from the general one for different values of parameters. The solutions of these are presented in terms of their moments and compared to the finite element approximation of the general case.

RICHARD EUBANKS, IBM Corp., Kingston, N.Y. A Statistical Mechanics Approach to Fore-

ground/Background Degradation. This paper introduces an analytical technique in analyzing foreground and background degradation

in a multi-task computer system. The approach utilizes the fundamentaf concepts of statistical mechanics, and in particular the statistical mechanics approach in analyzing queueing and network problems. The system environment consists of a low priority batch processing workload (background activity) and a real time higher priority foreground task. The partition function is used to clearly describe the multiplicity of system states and their respective probabilities. Consequently, this makes it possible to rigorously analyze complex system configurations and to consider the constraints of the operating system. Degradation is measured as the decrease in throughput of a particular task, and it is the



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contention between the tasks for a system resource which results in degradation. In order to depict the capability of mathematically treating a conversational system, the solutions of both a trivial and a complex system structure are derived.

TERRY A. FERRAR and ANDREW WHINSTON, Purdue University. The Cuefor a Private Environment Agency.

This paper explores the question of whether from society's point of view control of the environmental quality should remain in the private sector. Specifically, we propose a private agency that has defined for it public responsibility in that it must maintain certain environmental quality standards; moreover, this agency is given the right to sell user permits to potential municipal and industrial polluters or users of the environment. Furthermore, we specify the dual responsibility faced by the region's governing structure of both defining the environmental quality standards and controlling this private agency. The behavior of the regulated agency is examined as a constrained nonlinear optimization problem from which are derived reaction functions whose arguments comprise the quality standards and control parameters. The later sections of the work proceed to internalize the determination of the appropriate environmental quality specifications by the governing structure. This resulting nested optimization format is used to analyze the decision responsibility shouldered by the regional government which attempts to specify and protect environmental quality through a regulated private agency structure. This latter analysis is similar to that undertaken by Klevorick in relation to regulated industries [Bell J. Econ. Management, Spring (1971)].

HENRY E. FLEMING, National Oceanic and Atmospheric Administration, Washington. An Algorithm for the Approximate Solution of a Nonlinear Fredholm Integral Equation of the First Kind.

The integral form of the radiative transfer equation arising in connection with the remote sensing of atmospheric temperature profiles from satellites is

rb

g(x) = { K(x, y)F(x, f (y)) dy, a

where f(y) is to be determined and F has a known unique inverse. In practice g(x) is measured at a finite number N of points and the equation is reduced by numerical quadrature to a finite system of simultaneous equations. The algorithm is an iterative scheme given by

VN ~.2(X., yj gi - g(n)(xi)] M f(n+ l)(y.) = f(n)(yj) + El=, ' ( jJLb )/' = 1/ yj K2(x,y),

where it is assumed that the quadrature weights are included in K. Several examples are given to illustrate the method and the accuracies one can expect to achieve in practice.

HANS FOLLMER, Dartmouth College. The Potential Theory of Optimal Stopping and the "Bond Seller's Problem."

Let (X,) be a Markov process on a state space E and g a utility function defined on E. The optimal stopping problem consists in finding a stopping time Twhich maximizes the expected reward E[g(XT)]. We sketch the potential theoretic approach due to Snell, Dynkin, Shirjaev and others, removing some of the usual assumptions. This allows an application to the "bond seller's problem" introduced by W. M. Boyce: Stop a Brownian motion (Y1) starting in 0 and conditioned to some "predicted" distribution at a fixed deadline, say to a normal distribution N(m, U72) at time 1, so that the expected value E[YT] is maximized. Boyce uses a discrete approximation and numerical computation of the optimal strategy in the discrete case. We study directly the process (Y1), derive its stochastic differential equation and thus provide some insight into the qualitative phenomena appearing in Boyce's data, in particular the switch from a "sell on strong rallies, ride out storms" to a "cut losses, let profits run" strategy as the variance U2 passes the critical value 1.

HIROSHI Fujii, The University of Texas at Austin. Finite Element-Galerkin Method for Mixed Problems of Hyperbolic Equations.

Finite element-Galerkin schemes for approximating the solutions of mixed initial boundary value problems of hyperbolic equations of second order, and equations of linear elasticity are introduced. The main effort here is devoted to justify the finite element-Galerkin schemes of explicit type, of both



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consistent mass and lumped mass approximations. Continuous time, and implicit schemes are also considered. Stability conditions (sufficient conditions) which depend on the mesh ratio At/K, where At and K are the time increment and the minimum perpendicular length of all triangles which constitute approximate domain Qh, for the abovementioned schemes have been established. These conditions have been confirmed by numerical experiments. O(h) convergences in W")-spaces have been proved for the abovementioned schemes under appropriate assumptions.

MICHAEL GOLBERG, University of Nevada. An Initial Value Method for the Computation of the Eigenvalues and Eigenvectors of a Matrix A.

This paper presents a method for the computation of nonzero eigenvalues and eigenvectors of a real n x n matrix A by solving an initial value problem for n2 + 1 ordinary differential equations. The functions satisfying these equations are d(A) = det (I - AA) and D(i) = [Adj (I - ;A)]tA where Adj A is the adjoint matrix of A. The reciprocals of the zeros of d(i) give the eigenvalues and the columns of D(i), and its derivatives give the eigenvectors and generalized eigenvectors of A when A is a zero of d(A). The method is applied to set up a numerical method for computing the spectral properties of an integral operator. The paper also indicates how the Fredholm theory of integral equations can be viewed abstractly through a set of differential equations analogous to those for d(A) and D(i) above. Numerical results are given to demonstrate the feasibility of the method.

K. GOPSALMY, The University of Calgary. Doubly Stochastic Dynamic Systems and their Applica- tions.

Dynamic systems driven by white noise contaminated drivers have been considered by many authors. But for its mathematical simplicity, a white noise hypothesis is unrealistic in continuum mechanics. This paper proposes a new model of a dynamic system driven by a doubly stochastic process.

Suppose that the response u(x, t) of a linear dynamical system is representable (mean-square or w.p.l.) in the form

u(x, t) = n(s)G(x, t - s) ds, 0

where G(*) is the Green's function of the system and the driver n( * ) is made up of a sequence of pulses emitted at random times, each emitted pulse having a random intensity. When the emission process is Poissonian with an emission rate A(t), the probability density function p(z, t) of the response u(x, t) is shown to be governed by

ap/at + A(t)p = A(t) p(z - n(t)G(t), t) d1u,

p(Z, ) = b(z), -oo < z < oo,

where Q supports the probability measure ,u associated with random intensity n(t). If n(t) is a zero mean Gaussian process with a small variance o2(t), it is shown that

p(z, t) = {2it32(x, t)} -1/2 exp [-Z2/2/2(X,t)], -0 < z < 0,

where

/2(X t) = J(s)r2(s)G2(x, s) ds.

In terms of an easily computable "risk functional" the paper discusses the problems of modeling such phenomena as earthquakes, aerodynamic noise around busy airports and diffusion and decay of atmospheric pollution due to automobile and aircraft exhausts.

CHARLES A. HALL and JAMES KENNEDY, University of Pittsburgh. Ritz Approximations to a Problem of Elasticity.

R. Carlson and C. Hall [Ritz Approximations to two dimensional boundary value problems, Numer. Math., to appear] have investigated the convergence of various Ritz-Galerkin approximations to the plane strain elasticity problem for polygonal domains. The discretization error in approximating



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the components of displacement over the smooth Hermite space of order 2 was shown to be third order in terms of the L2-norm, as the mesh gauge h -+ 0. We establish that in fact the order of convergence is fourth order.

J. L. HESS, Douglas Aircraft Company, Long Beach, Calif. Numerical Solution of the Integral Equation for the Neumann Problem with Applications to Aircraft and Ships. (Invited.)

The problem of interest is that of incompressible potential flow about two- and three-dimensional bodies. Formulation of this problem as an integral equation over the body surface is an especially efficient procedure because the solution can be calculated on the body surface without considering the remainder of the flow field. The various possible integral equations that can be derived to represent the solution are discussed. Principal emphasis is on techniques of solution. The integral equation is replaced by a set of linear algebraic equations. The numerical problems are those associated with efficient calculation of the relevant "influence" matrices and with the solution of the resulting equations, whose coefficient matrix is nonsymmetric and full. Certain useful approximations are outlined, and direct and iterative matrix techniques are compared and evaluated. Alternate approaches pursued by other investigators are described. Sample calculations are presented, and some of them are compared with experiments to illustrate the remarkable agreement of potential flow and real flow. Finally, some unsolved problems of current interest are mentioned.

BERNARD E. HOWARD, University of Miami. Stabilization of Farm Prices. The concept of fixed price support has necessitated undesirable regulatory action (paying farmers

not to produce) to avoid over-abundance of supply. Apparently this is because the artificial support price is above the natural market price. This paper describes the concept of price-rate stabilization, in which governmental action is invoked only when price fluctuation is undesirably large, thus allowing private enterprise to operate to establish a natural market price with tolerable long range trends. The theory is based on the type of feedback control used to stabilize physical systems, such as guided missiles.

GEORGE C. HsIAo, University of Delaware, and RICHARD C. MACCAMY, Carnegie-Mellon University. On the Numerical Solution of a Singular Integral Equation Arising in Hydrodynamics.

In a previous paper by MacCamy [Arch. Rational Mech. Anal., 21 (1966), pp. 246-258] it was shown that the solution of the problem of Stokes flow past a cylinder can be reduced to the solution of a certain singular integral equation with a Cauchy-type singularity in the kernel. This paper concerns the numerical approximation for the solution of the singular integral equation subject to a constraint which is necessary for the uniqueness of the solution. The procedure is to replace the integral by a numerical quadrature, adjusted in a natural way to eliminate the singularity in the integrand, and then to solve a finite system of linear algebraic equations. In particular, a simple procedure is developed here for calculating the drag of the flow on cylinders of arbitrary shapes. Numerical examples for the case of ellipses of various eccentricities are presented and the results are discussed.

YASUHIKO IKEBE, The University of Texas at Austin. Numerical Computation of Mathieu Functions Using Integral Equations.

By Mathieu functions are meant even or odd periodic solutions with period X or 2i of the differential equation y" + (a - 2q cos 2x)y = 0, where q ? 0 is a prescribed parameter and a is a characteristic value, in order that the differential equation admit period solutions. In this work we make use of integral equations satisfied by the Mathieu functions in order to compute the coefficients of the Fourier series for each Mathieu function. For example, even Mathieu functions of period X satisfy the integral equation

4(?4(x) K(x, y)o dy, K(x, y) = cos (2k cos x cos y), k = q.

Letting 0(x) = Aoo0 + ' A21 42m(x), where 00 = (2X)- 1/2, n(x) - 1/2 cos (2mnx), m = 1, 2 one obtains an infinite matrix system (eigensystem) )z = Bz, where z = (A2m) and B = (Bpq),

Bpq = f { K(x,Y)Op(X)Oq(Y)dxdy, p, q =: 0 , 1,.



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This last matrix is real and symmetric. Each matrix element has the form

(const.) _ { K(x, y) cos (2px). cos (2qy) dx dy.

Fortunately, this last integral can be computed to be ( l)pn2Jp+q(k)Jp-q(k), p, q = 0, 1, . We solve this matrix system approximately by "truncating" B in an obvious way. We tried a number of cases including n = 8, 16, 32, 64 (n = the order of the truncated matrix) and q = 1, 22, 32, , 102,

202, 302, 402, 502 and obtained good results.

MARK M. JEPPSON, Colorado State University. A search for the fixed points of a continuous mapping.

Let cn be the unit n-cube and f an arbitrary continuous mapping from cn into cn. The algorithm described is a search for the fixed points off by locating "Sperner simplexes" in the n-cube. In obtaining the first Sperner simplex the method is analogous to those introduced by [H. Scarf, The approximation of fixed points of a continuous mapping, SIAM J. Appl. Math., 15 (1967), pp. 1328-1343] and [H. W. Kuhn, Simplicial approximation offixed points, Proc. Nat. Acad. Sci. U.S.A., 61 (1968), pp. 1238-1242]. The algorithm continues by disguising the original labels, exiting through an (n - 1)-face of the n-simplex and proceeding as before until coming to another "Sperner simplex" or a corner of the n-cube. The algorithm terminates when all possible exits have been made from all "Sperner simplexes." For x- in cn take 11x-1 = max(lx11, *- *, lxnl). For a given E > 0 there exists a ( > 0 such that

1f(x)- f (j) < E whenever L x- --1 < (. If x is in a "Sperner simplex" whose diameter is less than (, then L x- - f _( ? E + (. A computer program has been run on numerous examples.

LOUISE H. JONES, University of Delaware. On Finding Lower Boundsfor the Eigenvalues of Nonlinear Eigenvalue Problems.

Lower bounds for the eigenvalues of boundary value problems that are nonlinear in the eigenvalue parameter can be found from an equivalent Fredholm integral equation using a method developed by Goodwin [SIAM J. Appl. Math., 14 (1966), pp. 65-85]. This paper describes a method for finding lower bounds from an equivalent Volterra integral equation. It is shown that the two methods do not always produce the same lower bounds (an unexpected result in view of work by Brysk [J. Mathematical Phys., 4 (1963), pp. 1536-1538]) and that the lower bounds obtained by the Volterra method are generally superior to those obtained by the Fredholm method. Results of the application of the Volterra method to two nonlinear eigenvalue problems, the transverse vibration of a pipe containing flowing fluid and the transverse vibration of a rotating beam carrying tip mass, are given. Finally, a related method which can sometimes be used to determine whether the poles of the kernel of the integral equation are actually eigenvalues of the boundary value problem is discussed.

LOUISE H. JONES and R. E. KLEINMAN, University of Delaware. Numerical Solution of the Exterior

Neumann Problem for the Laplace and Helmholtz Equations.1 The problem of determining a solution of the Laplace or Helmholtz equation in the exterior of a

finite, smooth three-dimensional surface on which the values of the normal derivative are given is formulated as an integral equation for the unknown field on the surface. The integral equation differs from the standard equations derived from the Helmholtz representation or consideration of single and double layer surface distributions in that it is the continuous limit of a representation of the function in the exterior domain. The advantage of the new equation lies in the fact that the singularity in the kernel is cancelled, thus removing the major complication in other numerical treatments, and allowing a solution by straightforward iteration in a Neumann series. This iteration procedure has been programmed in Fortran IV for axially symmetric surfaces and results obtained for spheres and various cone spheres including the hemisphere. For small values of wave number in the Helmholtz equation (including the limiting case of Laplace equation) the iteration, even with primitive quadrature techniques, is found to converge very rapisdly.

R. E. KALABA, University of Southern California, Los Angeles. Initial- Value Methods for Non- linear Integral Equations with Applications in Biology. (Invited.)

1 This work was supported by the Air Force under Grant AFOSR 69-1794.



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It is shown that a wide class of nonlinear integral equations can be transformed into a Cauchy system. Then it is shown that a solution of the Cauchy system provides a solution of the original nonlinear integral equation. Such reductions are important because modern computers can solve initial value problems with speed and accuracy. There are intended applications in the theories of multiple scattering, optimal filtering, and lateral inhibition of neural systems. This new approach makes no use of successive approximations or series expansions.

W. J. KAMMERER, Georgia Institute of Technology, and M. Z. NASHED, Mathematics Research Center, University of Wisconsin. Iterative Methods for Best Approximate Solutions of Linear Integral Equations of the First and Second Kinds.

A matrix iterative method due to G. Cimmino is generalized to linear integral equations of the first kind: Kx = y. It is shown that the method converges monotonically to a function which minimizes the L2-norm of Kx - y and is of least L2-norm, starting from the initial approximation x0 = 0, for any y E R(K) + R(K)' provided that the dimension of R(K), the range of K, is greater than one. Using recent results of the authors [Steepest descent for singular operators with nonclosed range, Applicable Anal., 1 (1971); Oni the conuergence of the conjugate gradienit methodfor sinigular operator equatiols, SIAM J. Numer. Anal., to appear], the method of successive approximation, the steepest descent and the conjugate gradient methods are shown to converge to a least squares solution or to a least squares solution of minimal norm, both for integral equations of the first and second kinds.

W. J. KAMMERER, Georgia Institute of Technology, and G. W. REDDIEN, Vanderbilt University. Some Approximation Methods for Integral Equations.

Several approximation methods of projection type using polynomials and polynomial splines as approximating functions are given for Urysohn equations and Fredholm integral equations of the second kind. As an example, if the equation u(s) = f' K(s, t, u(t)) dt + g(s) has a solution u0 in CP[a, b], p > 1, with K 1, K3 and g continuous and a polynomial approximation u, of degree n is found by substituting u, into the equation and requiring the residual to be zero at the Gaussian nodes of degree n + 1 over [a, b], then the u, exist and are unique in some neighborhood of u0 for all n large, and lu - u0 ll0 = O(n-P+ 1/2). Other approximation methods and the results of numerical experiments are also presented.

H. G. KAPER, G. K. LEAF and A. J. LINDEMAN, Argonne National Laboratory. Formulation of a Galerkin-Type Procedure for the Approximate Solution of the Neutron Transport Equation.

The one-group neutron transport equation is commonly given as an integro-differential equation for the neutron flux V(x, co) over a domain Q = G x S in the five-dimensional phase space E3 x S, (Jo_) = 1). Neutron transport problems, which normally involve several media with different material properties, are conveniently formulated in a Sobolev-type space L j(Q). As was first shown by Vladimirov, it is possible to decompose the domain of the transport operator into a complementary pair of manifolds {M, N} by means of a projection operator P: P = '(I + U), where (U)(x, c)) = V(x, - c)). Any transport problem can then be reformulated on either manifold M or N, and each of these new formulations is equivalent to the original formulation. The two operators thus obtained have the same formal expres- sion they are again integro-differential operators, but involve second order derivatives with respect to the spatial variables. They are self-adjoint in the sense of Lagrange; a further reduction is necessary to render the boundary value problems self-adjoint. Using the theory of linear operators of elliptic type we look for weak solutions in the space L'(Q) and discuss a Galerkin-type approximation procedure.

RICHARD KRAFT, National Bureau of Standards. Contractivity of the Integral Equation of Inter- reflections.

The kernel of the integral equation describing radiation interreflections in a cavity is shown to define a contraction mapping.

PETER LINZ, University of California, Davis. The Numerical Solution of Dual Integral Equations. It is shown how Abel transforms can be used to convert dual integral equations into a single

integral equation of simple form. The resulting equation is treated by a numerical method, using piecewise polynomial approximations to the unknown function, together with product integration



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techniques. Some typical physical problems to which this method has been applied successfully are the computation of the capacitance of a circular plate condenser and the determination of the displacement in a penny-shaped crack. The technique can also be applied to systems of dual integral equations, such as that arising in the computation of the stresses in the indentation of an elastic medium by a rigid punch.

A. T. LONSETH, Oregon State University. Applications of Integral Equations. (Invited.) Integral equations come up in many areas of Applied Mathematics. In this paper no attempt will

be made to list all these areas, which would be tedious. However, it may be useful to describe some ways in which integral equations have arisen historically, and to point out features of the mathematical environment which favor their appearance. Examples will be mentioned from classical mechanics (dynamics, potential theory); from probability (renewal theory, stochastic processes and ergodic theorems); from the intersection of probability with mechanics (diffusion, kinetic theory of gases, radiative transfer, quantum theory); and from "additive heredity" (biological systems, magnetic and elastic heredity).

V. LOVASS-NAGY, D. L. POWERS and F. D. ULLMAN, Clarkson College of Technology. On Least- Squares Solutions of the Discretized Neumann Problem.

In the discretization of the Neumann problem V2u = -f in a region R and au/an = g on the boundary of R, one wishes to solve a problem of the form Mz = h, where M is a large singular square matrix. Even if the original problem satisfies the compatibility conditions, the discretized problem may not. In such a case, a reasonable compromise is the least-squares solution z = Mth, where Mt denotes the Moore-Penrose generalized inverse of M. Some methods are given for determining Mth in terms of nonsingular matrices, or matrices whose generalized inverses are easily found. (i) If M is an EPr matrix and PO(= PO) is the principal idempotent associated with the zero eigenvalue of M, then M + Po is nonsingular, and the generalized inverse of M is given by Mt = (M + Po) - Po. (ii) If M = H + V, where H and V are commuting, positive semidefinite matrices, then Mt = Ht(I + HtV)-1 + (I - HHt)Vt. (iii) If M = H + V, where H and V are commuting, positive semidefinite matrices, then the iterates defined by

(H + PI)Zk+ 1/2 = (PI - V)Zk + h, (V + PI)Yk+ 1 = (PI - H)Zk+ 1/2 + h, Zk+ 1 = (I - PO)Yk+ 1

converge to the least-squares solution of the equation Mz = h. (Po is the principal idempotent of M associated with the zero eigenvalue.)

TRILOK MANOCHA, IBM Corp., Kingston, N.Y. Ordered Motion for Direct-Access Devices. In computer systems, the seek-type storage devices service requests from the processing units on

a FIFO basis. This paper presents a mathematical analysis and simulation results of two new concepts of queue discipline. Each surface of the device is considered to consist of 200 concentric circles called cylinders on which blocks of data are stored. The queue consists of requests, where each request is for reading or writing a block of data on any cylinder with equal probability. A mechanical arm moves its read/write head hydraulically or electronically to the proper cylinder in order to service the request. We consider a constant queue, with a new member joining the queue when an existing member is serviced. The first discipline considered in this paper is such that the arm moves from the outer to the inner cylinders servicing requests along the way. When there is no request for data on a cylinder inner to its position, it moves back to the outermost cylinder for which data exists. It then continues inward. In the second discipline, the arm services requests in the direction described above as well as in the reverse direction. The practical advantage of these disciplines are described in the paper.

DAVID W. MATULA and RICHARD A. KOLDE, Washington University. Multiple Facilities Location- Allocation in Certain Sparse Networks.

Let N be an undirected network on n vertices with nonnegative distances on the edges. A k-median of N is a set of k of the n vertices chosen as locations for k identical facilities such that the sum of the distances of each vertex from its closest facility is minimized, and a k-center is a set of k out of n vertices chosen for facility location so as to minimize the maximum distance of any vertex from its closest facility. The determination of a k-median for an acyclic network is solved via a three-dimensional dynamic program where the computational effort needed to determine the solution is shown to have



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a growth rate of order kn2. For an acyclic network the determination of the minimum number of facilities needed so that no vertex is at a distance greater than d ? 0 is solved via a single pass labeling algorithm, and the efficient iterative utilization of this labeling procedure for determining a k-center is described. The extension of these techniques to those cyclic networks where distinct cycles have at most a single vertex in common is shown to be computationally viable.

T. W. McGUIRE and S. P. SETHI, Carnegie-Mellon University. Economic Dynamics with Hetero- geneous Labor.

In most control theoretic analyses of optimum economic growth the fraction of output allocated to investment is the control. The few models treating labor training assume innately identical individuals and a fixed amount of training for employability. We extend these models by assuming a CES production function of both skilled and unskilled labor (abstracting from capital) and a continuous distribution of innate ability. This results in a difficult continuous lag optimal control problem which can, however, be approximated by considering only a finite number of training classes as state variables. The control solution is obtained by the discrete maximum principle and is generally different from the market solution, assuming static expectations about future relative wages, except when the production function is linear. There exists, in general, a unique stationary optimal control not greater than the golden rule but equivalent to the market stationary control under certain circumstances. There also exists a stably controllable tax-subsidy system that induces dynamically optimal market behavior. Possibiiities of convergence to or chattering around the steady state are examined. Computer simulations of the optimal control and market dynamics are compared.

EARL H. McKINNEY, Ball State University. A Predictor-Corrector Method for First Order Differential Equations by Recursive Interpolation.

Recursive polynomial interpolation is used to derive a predictor polynomial (formula) over the set of data points (xi, y,) and derivatives y', i = 1, 2, 3, , n, which produces the Hermite interpolating polynomial over an arbitrary distribution of points. The corresponding corrector polynomial (formula) is obtained by recursive interpolation over the almost Hermite data set (xi, yi), i = 1, 2, 3, ..., n, and derivates of y', i = 2, 3, 4, ... , n, n + 1. The corrector polynomial is obtained from the predictor polynomial rather easily when the data points for recursive interpolation are arranged in the order (x2, Y2), (x2, Y'2), (X3,y3), (X3, y'3)... (Xn Y) (Xn Yn) (x1, Y), (x1, y'1) for the predictor and (x2, Y2), (x2, Y2), (X3, y3), (X3, y3)... (Xn Yn)) (XnY yn) (x1, Yi), (Xn+1 yYn+ ) for the corrector. Replace- ment of the final term in the predictor polynomial by the term produced by the data point (Xn+ 15 yYn+ 1) leads to the corrector polynomial. The scheme hence determines successively higher order P-C formulas as solution points are appended to the starting values and higher degree polynomials (P-C formulas) are obtained. Since the P-C polynomials are of the same order at each stage, error estimates are available. Optimum step sizes are determined assuming y(2n)(4) constant over the interval spanned by xi, i= 1, 2, .., n+ 1.

ORIN H. MERRILL, University of Michigan. Applications of an Algorithm that Computes Fixed Points of Convex Upper Semicontinuous Point-to-Set Mappings.

An algorithm developed by the author can, under certain conditions, find a fixed point of a nonempty convex upper semicontinuous point-to-set map F: Rn - Rn. This algorithm is similar to the ones developed by H. Scarf [The approximation of fixed points of continuous mappings, SIAM J. Appl. Math., 15 (1967), pp. 1328-1343] and B. C. Eaves [Computing Kakutanifixed points, Tech. Rep. 70-1, Dept. of Operations Research, Stanford University, Stanford, Calif.] A number of mathematical programming problems can be cast as fixed-point problems and solved using this algorithm. These applications include constrained and unconstrained minimization problems, the nonlinear comple- mentarity problem, the problem of finding a point in the intersection of a family of convex sets, and the problem of finding roots of systems of nonlinear equations. After each iteration of the algorithm, lower bounds on the value of the global minimum can be computed for some convex constrained and unconstrained minimization problems.

WEBB MILLER, Pennsylvania State University. On the Stability of Finite Numerical Procedures. By a finite numerical procedure we understand a numerical procedure which produces finitely

many numbers as outputs when given a finite set of parameters or inputs. Intuitively, such a procedure



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is stable if it is insensitive to roundoff errors. For a given set of parameters this sensitivity can be measured in two ways. One can either compare the computed answer with the exact answer, or, following Wilkinson, compare the set of parameters with those sets which would produce the computed answer assuming no roundoff error. Corresponding to each of these approaches are a number of precise definitions of stability. The purpose of this paper is to investigate relationships among some of these notions of stability.

J. A. MORRISON and J. McKENNA, Bell Telephone Laboratories, Inc., Murray Hill, N.J. Coupled Line Equations with Random Coupling.

The coupled line equations for two modes traveling in the same direction are considered, and the covariances of the mode transfer functions are calculated in the case of random coupling. Exact results are obtained by means of the Ito calculus when the coupling is white noise. Perturbation results are obtained in the case of weak, zero mean, wide sense stationary coupling, by application of a technique developed by G. C. Papanicolaou and J. B. Keller [Stochastic differential equations with applications to random harmonic oscillators and wave propagation in random media, SIAM J. Appl. Math., 21 (1971), pp. 287-305]. It is shown that perturbation results are valid in the case of strong coupling, if the correlation length is short. Exact results have also been obtained when the coupling is a function of a finite state Markov chain, by means of a procedure discussed earlier by J. A. Morrison [Moments and correlation functions of solutions of some stochastic matrix differential equations, SIAM National Meeting, Seattle, Washington, June 28-30, 1971].

ZANE C. MOTTELER, Gonzaga University. Towards an Understanding of Hydrodynamic Flow Through a Permeable Medium.

This paper considers a number of two-dimensional, steady-state flows of an incompressible fluid through a permeable medium, first with constant permeability, then with variable permeability. In all cases the permeability is a function of only one direction; we consider flows which are parallel, perpendicular, and oblique to this direction, as well as one more complicated flow. It is relatively easy to solve the resulting partial differential equation in each case, both for constant permeability and for variable permeability. By comparing the flows it is apparent that the main effect of raising the permeability is on the magnitude of the velocity, not upon its direction. Thus, lacking other external forces, a region of high permeability does not tend to "attract" fluid in its direction.

A. N. NETRAVALI, Optimal Data Corporation, Huntsville, Ala., and R. J. P. dE FIGUEIREDO,

Rice University. Spline Approximations to the Solutions of Fredholm and Volterra Integral Equations. Consider first the scalar (real) Fredholm equation

(1) (A - _t-)x = y,

where (.X?x)(t) = Sa K(t, s)x(s) ds, a ? t ? b. Note that if g E C'[a, b], then

[P 1 lIgIlp= sup E Z g()(t)0

a<t_b _x=O

Let Sa(f;) denote the ordinary cubic spline interpolating an appropriate function J: R' R' on a mesh An :a = to < t1 < < tn-1 < tn = b, and such that SXn(f;a) = f'(a) and SXn(f;b) = f'(b). Define the operator _tni by -tnf = _V-Sa(f; .). Denote by xn the solution of (A - i*n)xn = y, and by x,n the spline Sa (xn; .). The Anselone and Moore theory [J. Math. Anal. Appl., 9 (1964), pp. 268-277] is used to establish the following theorem.

THEOREM 1. Suppose that: (i) SUpa t<bba IK ()(t,s)I ds < M, < co, i = 0, 1; (ii) Sa IK"')(t, s) - K()(, s)I ds -O 0 as It - OI -- 0, t, t E [a, b]; (iii) (A - K - 1 exists; (iv) yeC'[a,b).

Then1 there exists an1 N such that for all ni _ N, (A- Xj- 1 exists anid, as n - oc, each of the sequenices lxn} an1d {IC} coniverges to the solution? x of(1) in the C'[a, b]-norm.

THEOREM 2. If K( ., s),for s E [a, b], and y belonig to C4[a, b], theni there exists an? N such thatfor all 1 > N, IIx - Xnllo< ? 11JAnill an1d lx - Xnio -< 32IA|nl|, where llAnll = maXk.l ln[Itk - tk 1/k, /2 positive conistanits.

Next consider the scalar Volterra equation

(2) (1 - A)x y,



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where (1':)(t) = fo IF(t, s)x(s) ds, 0 < t < c. Define the mesh Am = {O, h, 2h, , mh = t}. Let -jf = -*SA,(f; ), and Xj the solution of (1 - j)j = y. It is proved that, under appropriate conditions, lx-('(jh) - ?(jh)l < Ath4; M'! = positive constant, i = 0, 1, j = 1, * , m. Also, supo?t<mh[ISA(l (Xm; t) - .'0(t)I1 < Mh4-i, i = 0, 1, 2, where M = positive constant.

BEN NOBLE, University of Wisconsin. The Literature on Numerical Solution of Integral Equations, with Physical Applications. (Invited.)

An extensive bibliography on methods for solving integral equations is available in Reports 1176 and 1177 of the Mathematics Research Center. In spite of the fact that a considerable literature has appeared on the numerical solution of integral equations in recent years, it appears that algorithms are still relatively inefficient and a priori realistic error estimates are not available for complicated problems. There are few analyses of methods for solving singular integral equations, eigenvalue problems (bifurcation in the nonlinear case, and integral equations of the first kind, for example). In connection with physical applications, most work in the past has been devoted to finding approximate solutions by analytical methods. It would seem that numerical methods could be exploited much more in certain applications, for instance, mixed boundary value problems.

MICHAEL PAPADOPOULOS, Corvallis, Ore. Diffraction by a Dielectric Wedge. An infinite, homogeneous dielectric wedge scatters an incident plane wave of step-function

profile. The scattering involves not only the processes of reflection and refraction; there is also a diffraction effect associated with the corner of the wedge. The wedge angle is unrestricted; the angle of incidence is only limited in order to avoid the possibility of having totally reflected waves in the incident field.

The mathematical problem is to find the solution to a pair of linear wave equations, each having a distinct propagation constant, subject to linkage conditions at a couple of plane interfaces.

Here the solution is developed in the form of orthogonal series expansions. The coefficients are functions of position, determined by continuation from regions in which they are easily calculated into regions where they have not previously been found. This process is completely described, and details are given for the simple case of the 1200 wedge.

Of particular interest is the part played in the solution by a separation into regions for which the governing equations are either elliptic or hyperbolic; the determination of the linkage from a hyperbolic region into an adjacent elliptic region is sufficient for the determination of the solution.

JAMES L. PHILLIPS, Washington State University. Quadrature Techniques for Use with Projection Methods in Solving Linear Integral Equations.

Projection methods for solving operator equations view the problem of obtaining an approximate solution as a generalized curve-fitting problem. Usual formulations of such methods applied to linear integral equations ignore the practical problem of evaluating integrals which arise in applying the methods numerically. In this paper, quadrature techniques useful in applying projection methods to linear integral equations of the second kind are examined. We formulate conditions on the quadrature rule such that use of quadrature does not change the rate of convergence of the approximate solution. Error bounds for the resulting approximate solution are given. Specific examples are also discussed.

DONALD T. PIELE, The University of Wisconsin-Parkside, Kenosha. Asymptotically Neutral Families in E3.

Consider a bounded, open, connected region D in E3 with connected complement. For a sufficiently smooth Lyapunov boundary surface S, we construct an asymptotically nieutralfamily {Ynl, Yn2, a * * } Ynn, n = nj -oo, of points on S which, by definition, have the property that the sums of the potentials due to unit charges placed at {Yn,, Yn2, * Ynn} converge (modulo constants Cn) to zero as n = n;j - oo. Specifically, k (1/lix- Yn,k) + Cn -? 0 uniformly on every compact subset K c D. In the course of the construction we examine: (i) The equilibrium distribution ,u on 5, fS (u(y)/IIx - yll) du(y) = C, and how the Holder continuous differentiability of ,u is related to that of S; (ii) A proof of the strict positivity of ,u using a result of E. Hopf; (iii) An approximation to the integral fS (y(y)/1Ix - yll) du(y) by a sum of plane integrals each of which is further approximated by a Gauss type numerical integration rule. We conclude by relating asymptotically neutral families to the approximation of harmonic functions in D. The construction of asymptotically neutral families for bounded simply connected regions in E2 has been done by Jacob Korevaar [Ann. of Math., 80 (1964), pp. 403-410]. New techniques are developed in this paper to extend the results to En, n > 3.



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MICHAEL L. PODREBARAC and S. SANKAR SENGUPTA, University of Waterloo. On a Representation of Bargaining and Multi-Agency Action.

A complete distributive lattice is considered, equipped with a family of nonnegative real-valued homomorphisms which can be made a convex distributive lattice. Equipping the original lattice with a measure and the family of homomorphisms with a metric, the latter and its convex sets are seen to possess certain characteristic properties. The main result (Theorem 6) follows from a combination of these properties and a result due to Choquet. The mathematical scheme is interpreted in the customary terminology of "choice" among not necessarily quantity-measurable "alternatives." An example illustrates how the scheme can be employed to describe multi-group choices. In a second example, a class of "bargaining" situations is modeled in terms of linear programming with multiple objective functions combined with undetermined weights; the cost vectors in this formulation are identified with the homomorphisms, and Theorem 6 is applied.

W. G. POOLE, JR., College of William and Mary. An Initial Vector for Inverse Iteration in Eigen- vector Computations.

Given an eigenvalue A of a symmetric tridiagonal matrix T, inverse iteration is often used to find a corresponding eigenvector x,. We present here an initializing vector b, for inverse iteration whose components depend upon the leading principal minors of T - Ai I. This sequence of minors satisfies the Sturm sequence property and is calculated in a well-known method for finding the eigenvalues. b, is shown to be not pathologically deficient in x,, thereby guaranteeing fast convergence of inverse iteration. There appears to be no previously published technique for choosing a completely safe starting vector. The consequences of finite precision and numerical experiments using this technique are discussed.

D. A. PRELEWICZ, Washington University. Validity of Approximate Solutions for Nonlinear Boundary Value Problems.

Boundary value problems for equations of the type y" + f (y, y', t) = 0 are considered. Fixed- point theorems are used to establish the existence of an exact solution in a neighborhood of an approximate solution. An upper bound on the size of the neighborhood and local uniqueness are also obtained. Applications to problems arising in oscillation theory and nonlinear heat generation serve to illustrate the approach.

S. W. RAUCH, University of New Brunswick. A Convergence Theory for a Class of Nonlinear Programming Problems.

In this paper we study a nonlinear programming problem of the form

minimize {f (x): x E {y E D:gi(y) < 0, i = 1, *., m} ,

where fgi:D c R- R' are continuously differentiable on the open convex set D. In particular, we consider descent methods of the type

xk+ 1 = xk _ (kTk, f(xk) > f(xk+ ), k = 0, 1,

with a suitable choice of direction vectors Sk, step lengths Tk, and relaxation parameters (ok. A recent convergence theory of R. Elkin [Convergence theorems for Gauss-Seidel and other minimization algorithms, Doctoral dissertation, Univ. of Maryland, College Park, 1968] for such descent methods in the case of unconstrained minimization is extended to the above nonlinear programming problem. As in Elkin's original approach, the analysis of a variety of step length algorithms is treated entirely separately from that of several direction algorithms. This allows for their combination into many different methods for solving the constrained problem. We also extend the results of Topkis and Veinott to nonconvex sets and drop their requirement of the uniform feasibility of the search directions.

A. K. RAY and S. RANGACHARI, Calculation of First and Second Order Effects in the Theory for Glancing Interaction Between Shock Wave and a Laminar Boundary Layer Without Separation.

Glancing interaction between a shock wave and a laminar boundary layer with separation was solved with the help of the momentum balance equation [A. K. Ray and S. Rangachari, A theory for glancing interaction between a shock wave and a laminar boundary layer with separation, SIAM 1971 National Meeting, Seattle] . The case without separation was solved with the help of free stream velocities



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U x- and W = const. in the mainstream and cross-flow directions for small curvature of the external stream [A. K. Ray and S. Rangachari, Study of a similarity solution in a three-dimensional compressible laminar boundary layer, SIAM 1970 Fall Meeting, Boston]. In the present paper, the effects of curvature of the external stream have been investigated by perturbation analysis. The first order effects on the flow characteristics, namely skin friction and heat transfer, have been calculated by analytical methods involving integral representations of series solutions, slowly varying functions, and evaluated with the help of the method of steepest descent in terms of incomplete gamma functions; whereas the second order effects on them are obtained by solving the second order partial differential equations with series solutions.

JAMES A. RENEKE, Clemson University. A Variation of Parameters Formula. S is the number interval [0, T], {X, I l; is a complete normed Abelian group with zero element 0,

and M is the set of functions from X to X with identity function 1. Suppose that V1 and V2 are functions from S x S into M such that (i) V1 has its values in the homomorphisms of X, (ii) VP is order additive for each P in X and i = 1, 2, and (iii) there is an order additive function A from S x S to the numbers such that I V,(u, v)P - V,(u, v)Ql ? {(u, t)lP - Ql and I Vj(u, v)01 < A(u, v), for each (u, v) in S x S, (P, Q) in X x X, and i = 1, 2. The following equations are equivalent:

(1) h(u) = P + (R)f Vih + (R)f V2h

and

(2) h(u) = W1(u, O)P + (L, R) W1[u, ]V2h,

where

W1(u,v) = 1 + (R)f V1 W[.v]

for each (u, v) in S x S. Applications are made to integral equations and functional integral equations.

D. F. SANDERSON, Western Washington State College. On Numerically Stable Liniear Pr-ogramming Algorithms.

We report on the construction of a version of the simplex algorithm which is based on House- holder's QU decomposition for linear systems and compare it with Bartels' more efficient but possibly less stable algorithm which is based on the Gauss-Jordan L U decomposition [Doctoral thesis, Stanford University, Stanford, Calif., 1968]. Finally, we indicate results of comparative numerical experiments performed with these algorithms on the potentially very unstable L' polynomial regression problems.

P. E. SAYLOR, University of Illinois. Second Order Symmetric Factorization. A factorization method is a technique for solving an elliptic difference equation, Au = s, by con-

structing B such that (A + B)v = t is efficient to solve and then using this fact in an iteration to approximate u. An example of an effective factorization method is due to H. L. Stone [SIAM J. Numer. Anal., 5 (1968), pp. 530-558]. Stone constructs B in such a way that Bv = 0 when the components of v are the values over a system of gridpoints of a first degree polynomial in x and y. Accordingly, B may be said to be second order. Mathematical analysis of Stone's method is difficult for several reasons. One is that B is nonsymmetric. This motivates the need for a symmetric B. The principal result is that there is no second order symmetric B of any practical interest. Any symmetric B must be of lower order, of which the most promising is also due to Stone.

H. A. SCHENK, Naval Undersea Research and Development Center, San Diego, Calif. The Use of Integral Equation Techniques in Acoustic Radiation and Diffraction Problems. (Invited.)

Several numerical techniques for the solution of integral equations representing acoustic radiation and diffraction problems have been reduced to practice in recent years. A brief review of the principal techniques will be given. However, this paper will emphasize and illustrate recent, specific developments and improvements which extend the practical capabilities of these techniques. Examples will be given



CHRONICLE 539

of the marriage of these techniques with well-developed methods of structural analysis and with the electro-mechanical descriptions of transducers and transducer arrays. In the case of arrays of transducers for generating or receiving underwater sound, the complete electro-mechano-acoustical trans- duction process has been modeled. This allows prediction of electrical properties of the array such as input impedance and power; mechanical motion and stresses in the transducers; acoustic pressure fields at, near, and far from the array; and overall efficiency. The realization of these capabilities depends on minimizing matrix generation and solution time. A discussion of the many uses of symmetry and of several techniques for calculating the required integrals will be given.

MELVIN R. SCOTT, Sandia Laboratories, Albuquerque, and CHARLES W. MAYNARD, University of Wisconsin. A Relationship Between Green's Function and Invariant Imbedding.2

The superposition principle for boundary value problems associated with linear second order differential equations is usually described by introduction of a Green's function. Invariant imbedding has been employed to resolve the solution to problems of this type into contributions associated with the homogeneous problem with homogeneous boundary conditions, with inhomogeneous boundary conditions, and with an inhomogeneous term in the equation. The contribution to the solution from the inhomogeneous term is handled by two formalisms and these must be related.

For the equation

(1) w"(x) + a(x)w'(x) + b(x)w(x) = h(x), a < x < b,

and appropriate boundary conditions, the solution w(x) is related to the source h(x) by the Green's function g(x, x') through a standard superposition integral where g(x, x') satisfies (1) with a delta function source at x'. Invariant imbedding relates the solution and its derivative through

(2) w'(x) = S(x)w(x) + dx's(x, x')h(x')

and

(3) w(x) = R(x)w'(x) + dx'r(x, x')h(x'),

where the transformation functions S, s, R, and r satisfy a system of first order differential equations [C. W. Maynard and M. R. Scott, Invariant imbedding of linear partial differential equations via generalized Riccati transformations, Preprint SC-DC-70-5107, Sandia Laboratories, Albuquerque, 1970]. These are related to g(x, x') by

(4) s(x, x') dg(x, x')/dx - S(x)g(x, x')

and

(5) r(x, x') g(x, x') - R(x)dg(x, x')/dx.

C. B. SHAW, JR., North American Rockwell Science Center, Thousand Oaks, Calif. Generalization and Application of the Method of Best Accessible Estimation.3

Best accessible estimation is an iterative procedure based on the stochastic extension concept that an integral equation of the first kind is improperly posed because it ignores background noise on the unknown function, and measurement or computational error in the known function. Two restrictions on the method, introduced to combat extreme noise levels, have been removed. It is no longer required that the autocorrelation operator R for measurement error be diagonal, nor that the original equation be premultiplied by the adjoint operator. Except for an invariant component error in the "known" function, iterated estimation is now strictly norm-reducing as to residue in the ellipsoidal norm defined by R. Convergence (to the maximum-likelihood estimate, if measurement error is multivariate Gaussian) is thus assured in theory, and in practice accumulation of excessive numerical error can still be detected and iteration halted. The method is being used to enhance the effective resolution of energy distribution curves for semiconductor photoemission.

2 This work was supported by the U.S. Atomic Energy Commission. This work was supported by North American Rockwell I R & D Interdivisional Technology

Program, Mathematics and Statistics Project.



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J. LAURIE SNELL and DAVID S. GRIFFEATH, Dartmouth College. A Discrete Model for Stock Market Fluctuations Conditioned by Future Predictions.

The idea of using random walk or Brownian motion as a first approximation to stock market fluctuations is very old, probably first suggested by Bachelier in 1900. Recently Boyce and Shepp have discussed problems which arise from modifying the Brownian motion model to take into account future prediction information. Starting with the random walk model, we show how to modify the process to incorporate the new information. In terms of the concept of entropy (uncertainty), the original random walk model has maximum uncertainty. Our procedure is to modify the original random walk model by choosing a new probability measure which maximizes entropy consistent with the given prediction information. We are then able to explain, in terms of the modified model, some of the more striking qualitative features conjectured by Boyce in optimal selling and buying, given predictions of future prices.

Lucio TAVERNINI, Indiana University and Arizona State University. The Numerical Solution of Volterra Functional Differential Equations. (Invited.)

The purpose of this paper is to present a survey of numerical methods for the solution of Volterra functional differential equations, i.e., differential equations where the derivatives depend on the history of the solution. A large variety of retarded differential equations and of integro-differential equations are included as special cases.

The first half of the paper is about the development of a rather detailed theory of one-step and of multistep methods for the numerical solution of the real system

du(t)/dt = f (t, ut), t E [0, T], u(t) = v(t), t E [-z, O),

where u(t) c Rnv;v is a specified initial function; the function ut is "the history of u over the interval [t- , t]" and is defined by ut(s) = u(t + s) for - z ? s ? 0; and where f (t,.) is a functional satisfying certain continuity conditions.

In the second half of the paper some convergence results are given for the more general problem

du(t)/dt = Au(t) + f (t, tit), t E [0, T], u(t) = v(t), t E[--, O],

where u(t) E X, X is a real or complex Banach space; d/dt denotes the strong derivative; ut is defined as done above; and the operator A is linear, closed, densely defined on X, and is the infinitesimal generator of a continuous one-parameter semigroup of transformations fE(t)lt e [0, T]} on X. As before, the nonlinear term f (t, ut) depends on the history of u over [t - z, t] and satisfies certain continuity conditions.

Various examples and applications are included.

B. L. TURLINGTON, Southern Methodist University. Error Analysis of the Legendre Polynomial Method for Calculating the Laplace Transform.

The error analysis of a new method for the numerical calculation of the Laplace transform is presented in this paper. The method makes use of a new variable of integration so that the original limits of integration, zero and infinity, are transformed into other limits of integration, - 1 and 1. The function whose Laplace transform is desired is expanded in a series of Legendre polynomials. The use of the orthogonal properties of the Legendre polynomials and the interchange of integration and series summation signs allows the numerical calculation of the Laplace transform for real and positive values of the variable s. Only a finite sum of the coefficients of the Legendre polynomial expansion is required for any given approximation type. The method is particularly effective for small values of s. An error analysis is made for this method. Numerical approximations for various functions are made and the relation between the error bounds and actual errors is demonstrated.

B. L. TURLINGTON and D. K. DUNAWAY, Southern Methodist University. Some Major Modifi- cations to a New Methodfor Solving III-Conditioned Polynomials.

Several major modifications are made to a polynomial root finding method proposed by Garside, Jarratt and Mack. The modifications increase the computational efficiency by reducing the number of iterations and provide a set of starting values for the iterative procedure by implementing



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modified versions of Lehmer's method that assure convergence. The modifications are successful and a general purpose root finding algorithm that is at the present state of the art for solving ill- conditioned polynomials is produced. Numerical examples and comparisons are presented to sub- stantiate the modified method's effectiveness.

JOHN D. WALKER and S. SANKAR SENGUPTA, University of Waterloo. On a Representation of Utility.

An application of Riesz's representation theorem to represent the utility U(SA) of a collection (S) of goods, relative to a purpose (i), as a set function

(1) U(~~~~~~~~SJ)= f (4)M(d4), - R R,

is natural. It has the advantages that, for a given magnitude of U(Sj), it can be employed in (a) displaying pairwise compensating variations in f, ,u and S;, and (b) analyzing the deformations in the boundaries of S;. But the drawback is that the constructions cannot be easily interpreted. It is shown that the essential advantages can be retained if a certain modified form of the Radon-Lebesgue integral is viewed as a representation of linear functionals whose value at f ( ) is

(2) c Q(f) ff(u)M(d4), deR`, f(.)e C.

Examples illustrate the explicit and implicit determination of,u(.). The representation (2) is given the following interpretation: ;, a commodity-bundle; f( ), the offer-price function; Q(f), the income spent on S;. Consequently, Mi(dc) is interpreted as the maximum utility of "elementary" commodity sets.

DANIEL S. WATANABE, University of Illinois. The Numerical Solution of Volterra Integral Equations in the Kinetic Theory of Gases.

A class of initial boundary value problems governed by the integro-differential Krook model kinetic equation is reduced to an equivalent set of nonlinear Volterra integral equations in time. These equations differ from classical Volterra equations because their kernels contain integrals over space. A new implicit multistep method based on composite Gauss-Legendre quadrature anid Lagrange interpolation is given for solving these equations. The Gauss-Legendre quadrature rules are used instead of the usual Newton-Cotes rules because the kernels of these integral eqVations often vary rapidly. Numerical solutions of the nonlinear Rayleigh and piston problems obtained with this method are presented.

LEON E. WINSLOW, University of Notre Dame. The Numerical Solution of Ordinary Differential Equations by Finite Chebyshev Series.

Urabe [Numer. Math., 9 (1967), pp. 341-366] has given sufficient conditions to ensure that the solution of the differential equation y' = f(x, y) can be approximated arbitrarily closely by a finite Chebyshev series. An iterative method based on Chebyshev's acceleration method for calculating such finite approximations will be presented. The method is guaranteed to converge and, in fact, converges rapidly in practice. An error analysis, detailed examples, and a comparison to other methods will also be presented.

A. H. ZEMANIAN, State University of New York, Stony Brook. The Cascade Loading of Hilbert Ports.

A sufficient condition under which a Hilbert 2-port can be cascade loaded by a Hilbert 1-port is derived. It is assumed that the scattering operator of the 1-port is the convolution operator [W,k] where [w'Nk] is a 2 x 2 matrix of operator-valued distributions where supports are bounded on the left at the origin. It is also assumed that the 1-port is the convolution operatorf* wheref is also an operator- valued distribution with the same condition on its support. Cascade loading is feasible if the singularity of w22 is matched by the regularity off That is, if w22 is at most the nth derivative of a continuous operator-valued function and if f has continuous derivatives up to order n, then cascade loading has a sense.



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PHILIP B. ZWART, Washington University. Multivariate Splines with Nondegenerate Partitions. Any set of hyperplanes partitions En into a set of polyhedrons. An individual polyhedron is deter-

mined by selecting a half-space for each hyperplane and taking the intersection of these half-spaces. (Of course, some selections of half-spaces yield empty intersections.) A multivariate spline of degree n is a polynomial of total degree n on each polyhedron with all partial derivatives of order n - 1 being continuous everywhere. The hyperplanes are the generalization of the "knots" in the univariate spline theory. An especially simple canonical form is presented for splines with respect to nondegenerate (if a set of hyperplanes has nonempty intersection, then the corresponding set of normal vectors is linearly independent) partitions. Use of the canonical form, for fitting data, involves linear regression for fixed partitions and nonlinear regression for varying partitions. The canonical form gives rise to an ill-conditioned linear regression problem. However, some preliminary numerical experience in low dimensions indicates that the ill-conditioning is overcome with the use of singular value decomposition.

NORTHWEST SECTION

The SIAM Northwest Section held a regional meeting on November 13, 1971, at Washington State College, Ellensburg, Washington.

Titles and abstracts of papers presented are as follows: DAVID R. ANDERSON, Central Washington State College. Stokes' Theorem for Finite Sums. Difference Jorms are defined analogously to differential forms. Using partial differences rather

than partial derivatives, a difference operator d is defined which maps the (k - 1)-forms into the k-forms. The k-cells are copies of [0, 1]k embedded in n-dimensional space. Chains are formal sums of cells. A boundary operator a is defined. A sum (or "integral") of difference forms over chains is defined.

THEOREM. For any k-chain ai and any (k - 1)-form co,

,dco = , co.

SATYANADHAN ATLURI, University of Washington. Nonlinear Free Oscillation of Shells. The equations for the nonlinear free oscillations of a cylindrical shell are derived in the form of

two simultaneous nonlinear fourth order partial differential equations, one for the normal displacement of the shell midsurface and the other for the inplane stress function. A modal expansion is used for the normal displacement that satisfies the boundary conditions for the normal displacement exactly; but the boundary conditions for the inplane displacements are satisfied approximately by an averaging technique. The Galerkin variational technique is used to reduce the problem to a system of coupled nonlinear ordinary differential equations for the modal amplitudes. A multiple-time-scaling technique is used to solve these equations for arbitrary initial conditions; the amplitude-frequency relations of each mode and the general response are discussed.

DAVID C. BARNES, Washington State University. Bounds for the Eigenvalues of Sturm-Liouville Equations of 2nd Order.

A number of recent works have given eigenvalue estimates for Sturm-Liouville equations

( 1 ) U" + )P(X) U = 0, U(O) = U(a) = 0, where p(x) is smooth enough and positive. It is well known that an equation of form (1) may be transformed into an equation of the form

(2) ~~~~d2W (2) dt2 + ({-q(t))w = 0, w(O) = w(l) = 0.

It is not true, however, that an equation of form (2) may be transformed into an equation of form (1) since all eigenvalues of (1) are positive whereas the eigenvalues of (2) might be negative for certain functions q(t). The object of this work is to give eigenvalue estimates for (2) which correspond to those



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which are known for (1). For example, M. G. Krein has shown that if 0 < p(x) ? H and fo p(x) dx = m, then the nth eigenvalue in(p) of (1) satisfies the sharp inequalities

4Hn2 m 72H

m2 X aH < in(P)< n 2'

where X(t) is the smallest positive root of

(3) >X tan / = t/(1 - t).

This work, among other things, gives a sharp inequality of the form in(q) > A"(h, H, m, a), where in(q) is the nth eigenvalue of (2) and it is assumed that

a h ? q(x) < H, q(x)dx = m.

0

No sign restrictions are imposed on h and H, and the eigenvalues in(q) might be negative. The function An(h, H, m, a) is defined implicitly by a relation similar to (3).

ZANE C. MOTTELER, Gonzaga University. Towards an Understanding of Hydrodynamic Flow Through a Permeable Medium.

This paper considers a number of two-dimensional steady-state flows of an incompressible fluid through a permeable medium first with constant permeability, then with variable permeability. In all cases, the permeability is a function of only one direction. We consider flows which are parallel, perpendicular and oblique to this direction, as well as one more complicated flow. It is relatively easy to solve the resulting partial differential equation in each case, both for constant permeability and for variable permeability. By comparing the flows, it is apparent that the main effect of raising the permeability is on the magnitude of the velocity, not its direction. Thus, lacking other external forces, a region of high permeability does not tend to attract fluid in its direction.

SAM C. SAUNDERS, The Boeing Company. A Statistical Analysis of Stellar X-ray Sources. In this report we examine the statistical problems of locating a weak fluctuating radiation source

within a randomly varying natural background of obscuring radiation by using a sensor which scans a given area and integrates and records the total count for given periods of time. The assumption that a Poisson process governs the natural radiation at each fixed energy band is used to obtain a maximum likelihood estimate of the location of the object as well as a maximum likelihood estimate of the intensity of the source at that fixed energy band. Using a supposed location of the source, we can estimate the intensity at several energy bands in order to obtain an estimate of its spectrum. This technique is applied to some data taken from a search of the portion of the sky by such a sensor seeking stellar X-ray sources from a balloon. In this case a confounding of the data complicates the analysis because of the interruption of radio monitoring by antenna interference from shot being dropped to maintain a given altitude. A comparison of the accuracy of these statistical techniques is possible because of data obtained later from satellites and currently accepted location of such sources. These techniques can also be applied to location of a gamma radiation source within a prescribed area.

GEORGE W. SWAN, Washington State University. A Singular Eigenfunction Problem in Shock Wave Stability.

In a recent investigation into the stability of the permanent regime shock wave profile in a solid the following equation for a small perturbation quantity co(X, q) was determined:

a2w0 awo (X - X2)20 2 + 2(X - X2)@ -k = O,

0)(0, C) = 0)(1 C)= , where k is a positive constant; all quantities are without dimension. The solution of this problem is found to depend on the solution of a nontrivial singular eigenfunction problem with a continuous spectrum of positive eigenvalues distributed along the real line from 4 to oo. The quantity co(X, q) is finally represented as an inverse Laplace contour integral.



544 CHRONICLE

GABRIEL S. TSIANG, The Boeing Company. The Problem of Single and Double Reflection in Ray Tracing to find the Radar Cross Section of an Aircraft.

There are two ways of determining the radar cross section (RCS) of an aircraft using ray tracing method:

A. A geometrical optics ray tracing method involving single and double reflection of the rays; this method is employed for simple vehicle models.

B. If the aircraft is partitioned into many components, the aircraft RCS is obtained by two steps: 1. Determine the RCS of the components of the aircraft either by the method A or by direct

measurements. 2. Determine the RCS of the aircraft by the random phase method (which says that the RCS of the

aircraft is equal to the sum of RCS's at the given aspect angle for all the components that are not shadowed by other components). The method A has been published in IEEE Trans. Antenna and Propagation, September (1968); it is the intent of this paper to emphasize method B, which is primarily applied mathematics and computer oriented.

Single reflection. All components of the aircraft are simulated by ellipsoids (or portions of ellipsoids). Coordinates (t, u, w) are such that the direction of propagation becomes the negative w-direction. Therefore, backscattering from an ellipsoidal component would occur at the point where aw/au and aw/au are both equal to zero (i.e., the incident ray is normal to the ellipsoid at that point). An ellipsoid is determined to be unshadowed if the normal ray intersects it with a w-ordinate greater than all the other w-ordinates resulting from the intersection of the same ray with the remaining ellipsoids of the aircraft.

Double reflection. Aircraft components are simulated by ellipsoids and plane surfaces. An incident ray IF (in the negative w-direction) hitting a point F of a plane P. is reflected according to the rule: Angle of incidence equals angle of reflection. The reflected ray FS continues its path until it hits another component and is reflected according to the path SR. The path of the double reflected ray IFSR must have the characteristic: IF is parallel to and of the opposite sense as SR. Considerable testings must be performed to determine that the paths IF, FS, SR are not blocked by any other component so the random phase method can be applied for the determination of the monostatic RCS of the aircraft.

Summation of RCS. The RCS of the set S, of components detected to be unshadowed in single reflection and the set S2 of components (S, and S2 are mutually exclusive) detected in double reflection are summed to contribute to the RCS of the aircraft.

K. C. WANG, Washington State University. Decimation of Linear Binary Sequences from an Algebraic Point of View.

A linear binary sequence is a sequence of binary digits generated by a linear recurrence relation corresponding to a polynomial over the field GF(2). A decimation of the sequence is to select, in a prescribed order, any n bits at a time to form a sequence of n-bit words. The resulting sequence of words can be used as pseudorandom numbers or as sampling points in a statistical study.

This paper is concerned with the algebraic properties of the decimation schemes. For practical reasons, only maximum period sequences and decimations which preserve the sequence length are considered. Let the sequence of vectors in s' = As be called the basic sequence, where A is the companion matrix of the polynomial corresponding to a given recurrence relation. It is then shown that every decimation results in one of the following:

1. a sequence identical to the basic sequence with possibly a delay, 2. a sequence related to the basic sequence by a changing basis transformation, 3. a sequence corresponding to a new recurrence relation.

Conditions for each case to hold and its algebraic explanations are given in detail. Methods for determining the delay, the changing basis transformation, and the polynomial characterizing the new recurrence relation are also presented.

DAVID J. WOLLKIND, Washington State University. A Linear Stability Analysis of an Interface in an Isothermal Phase Transformation with Concentration Dependent Diffusion.

Diffusion phenomena have been of increasing importance of late. With this in mind and because it is the prototype of these problems dealing with the variation of material properties, a linear stability analysis of an interface in an isothermal phase transformation is carried out with the diffusion coefficient assumed to be a slowly varying function of concentration. The results of this analysis show that for the



CHRONICLE 545

precipitation system under consideration (one in which the distribution coefficient, a ratio of the equilibrium concentration of the solute on the solid side of the interface to that on the liquid side, is greater than one) there is a destabilizing influence if the diffusion coefficient is an increasing function of concentration and a stabilizing one if it is a decreasing function.

New Members DONALD JOHN ALBERS

Menlo College Menlo Park, California

MICHAEL ANTHONY ARBIB

University of Massachusetts Amherst, Massachusetts

RICHARD NORMAN BARSHINGER

Pennsylvania State University Dunmore, Pennsylvania

RONALD S. BASLAW

New York University Bronx, New York

PIERO BASSANINI

University of Perugia Perugia, Italy

ALAN M. BAUER

Bell Aerospace Company Buffalo, New York

GEORGES ANTOINE BECUS

State University of New York Buffalo, New York

DALE GAYLARD BETTIS

University of Texas at Austin Austin, Texas

PAUL ANTHONY BINDING

University of Calgary Calgary, Alberta, Canada

ROBERT P. CASTRO

Logicon Incorporated San Pedro, California

JOHN HOWELL CERUTTI

University of Wisconsin Madison, Wisconsin

KOK WAH CHANG

University of Calgary Calgary, Alberta, Canada

SUNDAY CHUKWUKA CHIKWENDU

University of California Los Angeles, California

WAI YUEN CHUM

University of Alberta Edmonton, Alberta, Canada

ROBERT BERNARD COOPER

Georgia Institute of Technology Atlanta, Georgia

JOHN KENNETH COOPER, JR.

Southern Colorado State College Pueblo, Colorado

KENNETH R. DRIESSEL

Amoco Production Company Tulsa, Oklahoma

IGOR JOHN EBERSTEIN

Goddard Space Flight Center Greenbelt, Maryland

BERNARD A. FLEISHMAN

Rensselaer Polytechnic Institute Troy, New York

SUDHANSHU KUMAR GHOSHAL

Jadovpur University Calcutta, India

FRED W. GLOVER

University of Colorado Boulder, Colorado

M. P. MARTIN GUZMAN

University De Madrid Madrid, Spain

DANIEL L. HANSEN

Northeastern State College Tahleguah, Oklahoma

JAMES R. HARGRAVE

Johnie Walker Medical Electronics Wichita Falls, Texas

AUBREY EATON HARVEY III Fayetteville, Arkansas

Jo ANN HOWELL

University of Texas Austin, Texas

GARY RoY JOHNSON

Itasca, Illinois AHMED LAKHDAR KEBAILI

*University of Tulsa Tulsa, Oklahoma

LAWRENCE ALFRED KURTZ

Hollins College Hollins College, Virginia

JOSEPH JAMES MALONE, JR.

Worcester Polytechnic Institute Worcester, Massachusetts

EUGENE ALEXANDER MARGERUM

United States Army Fort Belvoir, Virginia

DONALD LEON SEAN MCELWAIN

York University Ontario, Downsview, Canada

* Student Member.



546 CHRONICLE

RANDY SHERWOOD MCKNIGHT

Marathon Oil Company Littleton, Colorado

LEONARD JOSEPH MCPEEK

Malaspina College Nanaimo, British Columbia, Canada

RICHARD R. MITCHELL

Institute of Medical Sciences San Francisco, California

X. X. NEWHALL

*California Institute of Technology Pasadena, California

PAMELA JANICE PAJAS

Southfield, Michigan GORDON PALL

Louisiana State University Baton Rouge, Louisiana

YI-CHUAN PAN

Saint Mary's College WINONA, Minnesota

ALWAR PARTHASARATHY

*Syracuse University Syracuse, New York

LEONARD J. PUTNICK

Siena College Loudonville, New York

GERARD RUDOLPH RICHTER

University of California Los Angeles, California

* Student Member.

DAVID L. RiPPs Industrial Programming Incorporate New York, New York

RUSSELL E. SCHAUB

Super Valu Stores Incorporate Hopkins, Minnesota

CHARLES W. SCHELIN

Wisconsin State University LaCrosse, Wisconsin

SHOGI SHINODA

Tokyo, Japan JULIO E. SILVA

Berkeley, California K. S. SRA

Idaho State University Pocatello, Idaho

DONALD A. THOMPSON

C. F. Martin and Company Incorporate Nazareth, Pennsylvania

GABRIEL SHO-TSE TSIANG

The Boeing Company Seattle, Washington

MARGARET C. WAID

D. C. Teachers College Washington, District of Columbia

ROBERT L. WARNOCK

Illinois Institute of Technology Chicago, Illinois

DEATHS

The Editor regrets to announce the death of Professor STANLEY KATZ, of the City College of New York, in February 1972, a member of SIAM since 1953; and the death of Dr. GEORGE E. FORSYTHE, of Stanford University, in April 1972, a member of SIAM since 1956.



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